Pressure, temperature and density
Airspeed and altitude corrections begin with the atmosphere. Pressure, temperature and density change together, but they describe different properties.
The ISA reference
The International Standard Atmosphere provides the reference values used to calibrate instruments and compare aircraft performance. At mean sea level, ISA temperature is 15 degrees Celsius, pressure is 1013.25 hPa and density is 1.225 kilograms per cubic metre. In the troposphere, Bali uses a temperature lapse rate of 1.98 degrees Celsius per 1,000 ft to 36,090 ft. The temperature is then approximately minus 56.5 degrees Celsius through the lower stratosphere.
| Pressure altitude | ISA temperature | ISA pressure | ISA density |
|---|---|---|---|
| Mean sea level | 15 degrees C | 1013.25 hPa | 1.225 kg/m3 |
| FL 50 | 5 degrees C | 843 hPa | 1.056 kg/m3 |
| FL 100 | minus 5 degrees C | 697 hPa | 0.905 kg/m3 |
| FL 150 | minus 15 degrees C | 572 hPa | 0.771 kg/m3 |
| FL 200 | minus 25 degrees C | 466 hPa | 0.653 kg/m3 |
| FL 250 | minus 35 degrees C | 376 hPa | 0.549 kg/m3 |
Atmospheric layers used by the model
| Layer | Approximate height range | Temperature pattern in Bali's outline |
|---|---|---|
| Troposphere | Surface to about 11 km or 36,090 ft | Falls about 1.98 degrees Celsius per 1,000 ft |
| Tropopause | Near 11 km | About minus 56.5 degrees Celsius |
| Stratosphere | Above the tropopause to about 50 km | Initially nearly constant, then increasing |
| Mesosphere | About 50 to 80 km | Decreases with height |
| Thermosphere | Above about 80 km | Increases with height |
Pressure altitude
Pressure altitude is the altitude indicated when the altimeter subscale is set to 1013.25 hPa. It represents height in the ISA pressure structure, not necessarily geometric height above mean sea level. Flight levels are pressure-altitude surfaces. Pressure altitude is also the reference input used with temperature to determine density altitude and TAS.
| Setting or value | Meaning | Indication at the datum |
|---|---|---|
| QFE | Pressure at the aerodrome datum or runway threshold datum. | Approximately zero height |
| QNH | QFE reduced to mean sea level using the standard atmosphere. | Aerodrome elevation |
| 1013.25 hPa | Standard setting used for pressure altitude and flight levels. | QNE is the indicated value on the ground |
| QFF | Pressure reduced to mean sea level using the actual atmospheric conditions. | Meteorological value, not an altimeter setting for flight |
Pressure spacing with height
Equal pressure changes do not represent equal height changes. Bali gives about 27 ft per hPa at mean sea level, 34 ft per hPa at 10,000 ft, 50 ft per hPa at 18,000 ft and 100 ft per hPa near 40,000 ft. The growing spacing reflects the reduction of pressure with altitude.
How density responds
Higher pressure increases density. Higher temperature reduces density because the air expands. Water vapour is lighter than the dry-air molecules it replaces, so increasing humidity also reduces density. With increasing altitude the pressure effect normally dominates, therefore density decreases.
The airspeed correction chain
IAS, CAS, EAS and TAS are successive descriptions of one flight condition. Each correction removes a different source of error.
From indication to motion through the air
| Stage | How it is obtained | What it represents |
|---|---|---|
| ASIR | The raw airspeed-indicator reading before corrections. | Instrument display value |
| IAS | ASIR corrected for instrument error. | Indicated airspeed |
| CAS | IAS corrected for position or pressure error. | Calibrated airspeed; older texts may call it RAS |
| EAS | CAS corrected for compressibility error. | Equivalent airspeed |
| TAS | EAS corrected for density error. | Actual speed through the undisturbed air mass |
Instrument and position errors
Instrument error arises from imperfections in the indicator and its mechanism. Position error arises because the static source may not sense undisturbed ambient pressure at every speed, configuration and angle of attack. In practice, an aircraft correction card often combines both corrections so the pilot obtains CAS directly from IAS.
Compressibility error
The air entering the pitot system is compressed. As speed and altitude increase, this effect becomes significant. Removing compressibility error from CAS gives EAS. Oxford's training convention treats the correction as insignificant when the first TAS estimate is 300 kt or less, but it must be considered above that value.
In Oxford's worked high-speed example, CAS is 280 kt at FL 350 with SAT minus 47 degrees Celsius. Density correction alone first suggests about 500 kt TAS, which exceeds the 300 kt training threshold. Applying the compressibility correction reduces the result to approximately 480 kt TAS.
What each speed is used for
IAS is immediately available to the pilot and relates closely to aerodynamic loading. CAS removes installation error. EAS is the useful aerodynamic comparison after compressibility correction. TAS is required for navigation because it is the magnitude of the aircraft's air vector.
Instrument and position corrections change indicated airspeed to calibrated airspeed.Oxford ATPL Book 10, chapter 6
Correct CAS for compressibility to obtain EAS, then correct EAS for density to obtain TAS.R.K. Bali, Air Navigation, chapter 11
Density error and true airspeed
The ASI responds to dynamic pressure. When density changes, the same TAS does not produce the same indication.
The density correction
Relative density is actual density divided by ISA mean sea-level density. At lower density, the denominator is smaller, so TAS is higher than CAS. At greater density, TAS moves closer to CAS.
Worked ISA example
At FL 200 in ISA, density is 0.653 kilograms per cubic metre. Divide this by 1.225 to obtain a relative density of 0.533. Its square root is 0.730. A CAS of 100 kt therefore gives 100 divided by 0.730, which is approximately 137 kt TAS.
| Change at constant CAS | Density response | TAS response |
|---|---|---|
| Pressure altitude increases | Density decreases | TAS increases |
| Temperature increases | Density decreases | TAS increases |
| Humidity increases | Density decreases slightly | TAS increases slightly |
| Pressure altitude decreases | Density increases | TAS decreases towards CAS |
Quick estimates
A useful low-altitude planning estimate is that TAS increases by about 2 percent of CAS for each 1,000 ft of pressure altitude. Bali also gives the approximation TAS equals RAS plus 1.75 multiplied by RAS multiplied by flight level, divided by 1,000. These rules ignore detailed temperature and compressibility effects and are suitable only when the required accuracy permits.
Static and total air temperature
A temperature probe on a moving aircraft senses air that has been slowed and compressed. The indicated total temperature is therefore warmer than the undisturbed air.
SAT, TAT and ram rise
Static air temperature, also called outside air temperature, is the ambient temperature of the undisturbed air mass. Total air temperature includes the heating produced when airflow is brought towards rest at the probe. The difference TAT minus SAT is ram rise. Indicated air temperature and ram air temperature are older labels commonly associated with the total-temperature indication.
Recovery factor
A real probe may not recover the full theoretical ram rise. Recovery factor R describes the fraction recovered. A perfect probe has R equal to 1. The calculation must use absolute temperature in kelvin.
At FL 310, TAT is minus 9 degrees Celsius, Mach number is 0.86 and R is 1. Convert TAT to 264 K. Divide by 1 plus 0.2 multiplied by 0.86 squared to obtain about 230 K, or minus 43 degrees Celsius SAT.
A TAS form of ram rise
At 400 kt TAS, ram rise is approximately 400 divided by 87.1 squared, about 21 degrees Celsius. A SAT of minus 23 degrees Celsius would therefore produce a TAT near minus 2 degrees Celsius with full recovery.
Local speed of sound and Mach number
Mach number compares TAS with the local speed of sound. Temperature, not pressure altitude by itself, controls that local sound speed.
Local speed of sound
At ISA mean sea level, temperature is 288.15 K. The formula gives about 661 kt. Colder air has a lower local speed of sound; warmer air has a higher one.
Mach number
A Mach meter combines pressure information internally, so the displayed Mach number does not need a separate pilot-applied compressibility correction. The correction still matters when converting an indicated or calibrated airspeed through the complete airspeed chain.
If SAT is minus 42 degrees Celsius, absolute temperature is 231 K. Local speed of sound is about 593 kt. At Mach 0.86, TAS is 0.86 multiplied by 593, approximately 510 kt.
| SAT | Kelvin | Approximate LSS | TAS at Mach 0.80 |
|---|---|---|---|
| 15 degrees C | 288 K | 661 kt | 529 kt |
| minus 25 degrees C | 248 K | 614 kt | 491 kt |
| minus 42 degrees C | 231 K | 593 kt | 474 kt |
| minus 55 degrees C | 218 K | 575 kt | 460 kt |
Worked high-level check
At FL 350, Mach 0.80 and SAT minus 55 degrees Celsius, the local speed of sound is about 575 kt and TAS is about 460 kt. No mechanical navigation-computer instruction is needed: the relationship follows directly from temperature and Mach number.
How the speeds change with altitude
The direction of change depends on which speed is held constant and on whether temperature falls, stays constant or rises with altitude.
Climbing through ISA conditions
| Held constant | CAS or IAS | TAS | Mach | LSS |
|---|---|---|---|---|
| CAS | Constant | Increases | Increases | Decreases |
| TAS | Decreases | Constant | Increases | Decreases |
| Mach | Decreases | Decreases | Constant | Decreases |
Pressure and density fall as the aircraft climbs. In the ISA troposphere, temperature and LSS also fall. Thus, at constant TAS the Mach number rises; at constant Mach the TAS falls.
Isothermal layer
Temperature and LSS remain constant with height. At constant Mach, TAS also remains constant while CAS falls during a climb. At constant CAS, TAS and Mach both increase as density falls. At constant TAS, Mach remains constant while CAS falls.
Inversion layer
Temperature and LSS increase with height. At constant Mach in a climb, TAS increases. At constant TAS, Mach decreases. At constant CAS, falling density tends to increase TAS, while the warmer temperature also increases LSS; the resulting Mach trend must be assessed from both changes.
One TAS at several heights
For a TAS of 555 kt, ISA local sound speed is about 662 kt at sea level, 638 kt at 10,000 ft and 614 kt at 20,000 ft. The corresponding Mach numbers are approximately 0.84, 0.87 and 0.90. TAS has not changed, but Mach has risen because LSS has fallen.
| Layer in a climb | Temperature | LSS | At constant Mach |
|---|---|---|---|
| ISA troposphere | Decreases | Decreases | TAS decreases |
| Isothermal | Constant | Constant | TAS remains constant |
| Inversion | Increases | Increases | TAS increases |
Density altitude
Density altitude expresses actual density as the altitude at which that density would occur in ISA. It is a performance index, not a measured vertical distance.
Definition and effect
Start with pressure altitude, then account for the departure of actual temperature from ISA. Hotter-than-ISA air is less dense, so density altitude is higher than pressure altitude. Colder-than-ISA air is denser, so density altitude is lower. High density altitude reduces engine, propeller, rotor and wing performance for a given indicated condition.
Oxford rounds the factor to 120 ft per degree Celsius for mental calculation. Bali's 118 factor is the source-of-record value for this course.
Worked field example
Pressure altitude is 5,000 ft and OAT is 25 degrees Celsius. ISA temperature is 15 minus 5 multiplied by 1.98, which is 5.1 degrees Celsius. The deviation is therefore plus 19.9 degrees Celsius. Density altitude equals 5,000 plus 118 multiplied by 19.9, approximately 7,348 ft.
Oxford's rounded example uses pressure altitude 5,500 ft and SAT 35 degrees Celsius. ISA temperature is about 4 degrees Celsius, so deviation is plus 31 degrees. Using Oxford's 120 factor gives 9,220 ft. Using Bali's source-of-record factor of 118 gives approximately 9,158 ft, which is the preferred arithmetic when answer choices distinguish the methods.
Do not confuse the altitude terms
| Term | Reference | Main use |
|---|---|---|
| Indicated altitude | Altimeter indication on the selected setting | Operational vertical position |
| Pressure altitude | ISA pressure surface with 1013.25 hPa set | Flight levels and calculation input |
| True altitude | Actual vertical distance above mean sea level | Terrain and obstacle clearance |
| Density altitude | ISA altitude with the same density as the actual air | Aircraft performance |
Using the values in navigation
A navigation solution is only as sound as its inputs. Pressure altitude and SAT lead to TAS; TAS then becomes the air-vector magnitude in the triangle of velocities.
The practical sequence
- Read IAS and apply the approved aircraft corrections to obtain CAS.
- Apply compressibility correction when significant to obtain EAS.
- Use pressure altitude and SAT to apply density correction and obtain TAS.
- Use TAS with heading to form the air vector.
- Add the forecast or measured wind vector to obtain track and groundspeed.
- Use groundspeed, not TAS, for time over a ground distance.
Why TAS belongs in the velocity triangle
Wind is the motion of the air mass over the ground. The aircraft's motion relative to that moving air mass is TAS on the heading. Adding those two motions gives the ground vector. IAS, CAS and EAS are valuable aerodynamic quantities, but none is the speed of the aircraft through the air mass required by the vector equation.
Air data computers
An air data computer receives static pressure, total pressure and total air temperature, with system inputs such as electrical power and, where fitted, angle of attack. It can provide altitude, IAS, Mach number, TAS, SAT and density information. The displayed result still depends on serviceable sensors, valid source selection and correct interpretation.
| Given | Find | Relationship |
|---|---|---|
| CAS, pressure altitude, SAT | TAS | Correct for compressibility as required, then density |
| Mach number and SAT | TAS | Find LSS from kelvin temperature, then multiply by Mach |
| TAT, Mach and recovery factor | SAT | Remove ram rise using the absolute-temperature formula |
| Pressure altitude and ISA deviation | Density altitude | Add 118 ft per degree Celsius of deviation |
| Heading, TAS and wind | Track and groundspeed | Close the triangle of velocities |
Reasonableness checks
- At altitude, TAS is normally greater than CAS for the same indicated condition.
- TAT must not be colder than SAT when ram rise is positive.
- At a fixed Mach number, colder air gives lower TAS.
- Hotter-than-ISA conditions give density altitude above pressure altitude.
- At ISA mean sea level, LSS should be close to 661 kt.